In probability theory, the Schramm–Loewner evolution with parameter κ, also known as stochastic Loewner evolution (SLEκ), is a family of random planar curves that have been proven to be the scaling limit of a variety of two-dimensional lattice models in statistical mechanics. Given a parameter κ and a domain U in the complex plane, it gives a family of random curves in U, with κ controlling how much the curve turns. There are two main variants of SLE, chordal SLE which gives a family of random curves from two fixed boundary points, and radial SLE, which gives a family of random curves from a fixed boundary point to a fixed interior point. These curves are defined to satisfy conformal invariance and a domain Markov property. It was discovered by Oded Schramm (2000) as a conjectured scaling limit of the planar uniform spanning tree (UST) and the planar loop-erased random walk (LERW) probabilistic processes, and developed by him together with Greg Lawler and Wendelin Werner in a series of joint papers. Besides UST and LERW, the Schramm–Loewner evolution is conjectured or proven to describe the scaling limit of various stochastic processes in the plane, such as critical percolation, the critical Ising model, the double-dimer model, self-avoiding walks, and other critical statistical mechanics models that exhibit conformal invariance. The SLE curves are the scaling limits of interfaces and other non-self-intersecting random curves in these models. The main idea is that the conformal invariance and a certain Markov property inherent in such stochastic processes together make it possible to encode these planar curves into a one-dimensional Brownian motion running on the boundary of the domain (the driving function in Loewner's differential equation). This way, many important questions about the planar models can be translated into exercises in Itô calculus. Indeed, several mathematically non-rigorous predictions made by physicists using conformal field theory have been proven using this strategy.
The Loewner equation
If D {\displaystyle D} is a simply connected, open complex domain not equal to C {\displaystyle \mathbb {C} } , and γ {\displaystyle \gamma } is a simple curve in D {\displaystyle D} starting on the boundary (a continuous function with γ ( 0 ) {\displaystyle \gamma (0)} on the boundary of D {\displaystyle D} and γ ( ( 0 , ∞ ) ) {\displaystyle \gamma ((0,\infty ))} a subset of D {\displaystyle D} ), then for each t ≥ 0 {\displaystyle t\geq 0} , the complement D t = D ∖ γ ( [ 0 , t ] ) {\displaystyle D_{t}=D\smallsetminus \gamma ([0,t])}
of γ ( [ 0 , t ] ) {\displaystyle \gamma ([0,t])} is simply connected and therefore conformally isomorphic to D {\displaystyle D} by the Riemann mapping theorem. If f t {\displaystyle f_{t}} is a suitable normalized isomorphism from D {\displaystyle D} to D t {\displaystyle D_{t}} , then it satisfies a differential equation found by Loewner (1923, p. 121) in his work on the Bieberbach conjecture. Sometimes it is more convenient to use the inverse function g t {\displaystyle g_{t}} of f t {\displaystyle f_{t}} , which is a conformal mapping from D t {\displaystyle D_{t}} to D {\displaystyle D} . In Loewner's equation, z ∈ D {\displaystyle z\in D} , t ≥ 0 {\displaystyle t\geq 0} , and the boundary values at time t = 0 {\displaystyle t=0} are f 0 ( z ) = z {\displaystyle f_{0}(z)=z} or
g 0 ( z ) = z {\displaystyle g_{0}(z)=z} . The equation depends on a driving function ζ ( t ) {\displaystyle \zeta (t)} taking values in the boundary of D {\displaystyle D} . If D {\displaystyle D} is the unit disk and the curve γ {\displaystyle \gamma } is parameterized by "capacity", then Loewner's equation is
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